Classical Mechanics

Question 1

Define the Frictional force.

Answer 1

F_f = μF_N, where F_N is the normal force and μ is the frictional coefficient.

Question 2

Give the x- and y- equations of motions for a projectile in motion.

Answer 2

x(t) = v_0xt + x₀ and y(t) = -½gt² + v_0yt + y₀

Question 3

Give a formula relating the initial and final velocities of an object, its acceleration and the change in position between the initial and final states, if acceleration is constant.

Answer 3

v²_f - v²_i = 2aΔy.

Question 4

In terms of circular motion, if its tangential acceleration is zero, then its tangential velocity is constant; it is moving in uniform circular motion about the center of the circle. Give the radial acceleration and the centripetal force.

Answer 4

a = v²/r , F_cent. = mv²/r

Question 5

State the concept of conservation of energy.

Answer 5

If an object acted on only conservation forces, the sum of its kinetic and potential energies is constant along the object’s path.

Question 6

What are conservative forces?

Answer 6

A force to which you can associate a (time-independent) potential energy.

The work done by these forces is independent of the path taken between the starting and ending points.

Question 7

General principle:

If you want to know how fast or how far someting goes, use _____________________.

If you want to know how much time something takes, use _______________.

Answer 7

Conservation of Energy.

Kinematics.

Question 8

Give the formula of translational kinetic energy.

Answer 8

T = ½mv².

Question 9

Give the formula of rotational kinetic energy.

Answer 9

KE_rotational = ½Iω².

Question 10

Give the formula for Gravitational potential energy on Earth.

Answer 10

U_grav. = mgh

Question 11

Give the formula of the potential energy of a spring.

Answer 11

U_spring = ½kx².

Question 12

For any conservative force F, the change in potential energy ΔU between a and b is

Answer 12

ΔU = - ∫ F·dl, where the integral goes from a to b.

Question 13

Give the gravitational force between two masses m₁ and m₂.

Answer 13

F_grav = Gm₁m₂/r² r^.

Question 14

Give a alternate formula of ΔU = - ∫ F·dl.

Answer 14

F = -∇U.

Question 15

If the object rolls without slipping, then its linear velocity and angular velocity are related how?

Answer 15

v = Rω.

Question 16

Give a formula of work, due to non conservative forces, and energy.

Answer 16

E_i + W_{non conservative} = E_f.

Question 17

Give the work energy theorem.

Answer 17

W_conservative = ΔKE

Question 18

Give the general defintion of work.

Answer 18

W = ∫ F·dl.

Question 19

What can be applied if F_ext = 0 in a system?

Answer 19

F_ext = ṗ, by Newton’s second law.

Momentum is conserved.

Question 20

If things are colliding, try ____________________________ first.

Answer 20

Conservation of Momentum.

Question 21

The angular momentum of a point particle of linear momentum is defined by

Answer 21

L = r x p.

Question 22

The angular momentum of a extended body of linear momentum is defined by

Answer 22

L = Iw.

Question 23

The analogue of the force F for rotational motion is

Answer 23

Torque.

τ = r x F

Question 24

The analogues of the equations p = mv and F = dp/dt, in scalar forms are

Answer 24

L = Iw and τ = dL/dt.

Question 25

The angular momentum vector L is generally parallel to the ________________, which points along the axis of rotation, just like an object’s linear momentum is parallel to its velocity.

Answer 25

angular velocity w

Question 26

Is a reference frame rotating at constant angular velocity inertial or not?

Answer 26

Not, but one can still write a formula resembling Newton’s second law at the price of introducing “fictitious” forces.

Question 27

In a rotating frame, this force is responsible for the deviaiton of true g in a non inertial reference frame.

Answer 27

The centrifugal force.

F_centrifugal = -mΩ²r.

Question 28

In a rotating frame, this force is responsible for the curvature trajectory of bullets and hurricanes.

Answer 28

The Coriolis force.

F_Coriolis = -2mΩ x v.

Question 29

Define the moment of inertia of a point particle of mass m.

Answer 29

I = mr².

Question 30

Define the moment of interia of a rigid body.

Answer 30

I = ∫ r² dm.

Question 31

Conceptually, objects with more mass further from the axis of rotation are _______ to rotate and have a _____ moment of interia.

Answer 31

“harder”, larger.

Question 32

Define the parallel axis theorem which is used for moments of interia.

Answer 32

I = I_cm + Mr².

Question 33

Define the center of mass of an extended object of mass M.

Answer 33

r_CM = ∫ r dm / M

Question 34

Define the center of mass formula for point particles.

Answer 34

r_CM = Σ_i r_im_i / M

Question 35

Define the general Lagrangian formula of a system.

Answer 35

L(q,q̇,t) = T - U.

Question 36

Define the general Euler - Lagrange equations.

Answer 36

d/dt (∂L/∂q̇) = ∂L/∂q.

Question 37

Define the momentum conjugate to q.

Answer 37

p_i ≡ ∂L/∂q̇

Question 38

Iff the Lagrangian is independent of a coordinate q, the corresponding conjugate momentum ∂L/∂q̇ is what?

Answer 38

conserved.

d/dt(∂L/∂q̇) = 0

∂L/∂q̇ is constant.

Question 39

If U does not depend explicitly on velocities or time, define the Hamiltonian.

Answer 39

H = T + U.

Question 40

Give the general definition of the Hamiltonian.

Answer 40

H(p,q) = Σp_iq̇_i - L

Question 41

Give Hamilton’s equations.

Answer 41

ṗ = - ∂H/∂q, q̇ = ∂H/∂p

Question 42

Iff the Hamiltonian is independent of a coordinate q, the corresponding conjugate momentum p is what?

Answer 42

conserved.

ṗ = 0.